The main problem here is that the only physical things (ignoring stuff like half life, and so forth) you can measure of a particle are mass and charge. An electron is indivisible; once you know the mass and the charge, you know everything about a generic electron that you can possibly know. There isn't some finer structure, it isn't made of anything else, it just is. We can't even be sure it has volume in the conventional sense, aside from its probability shell. The only thing particles are made of are other, smaller, particles, and there's only so far this goes on for before you hit the bottom of the well; they're not like rocks, you can't cut them up and see what they're like inside; they're the way they are because... they're the way they are.
Addendum:
Protons and neutrons (and all hadrons) are made up of Quarks, but so far quarks have been found to be indivisible, so that's where the buck stops for the moment.